Lecture 6. Many-Electron Atoms. Pt.4.Physical significance of Hartree-Fock

solutions:Electron correlation, Aufbau principle, Koopmans’ theorem & Periodic trends

References

• Ratner Ch. 9.5-, Engel Ch. 10.5-, Pilar Ch. 10• Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 3.3 • Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7• Computational Chemistry, Lewars (2003), Ch. 5

• A Brief Review of Elementary Quantum Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.htmlhttp://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html

• Electron-electron repulsion• Indistinguishability

Helium Atom First (1 nucleus + 2 electrons) (Review)

We cannot solve this Schrödinger equation analytically. (Two electrons are not separable nor independent any more.)

A series of approximations will be introduced.

1. Electron-electron repulsion (correlation)

The 1/r12 term removes the spherical symmetry in He.

~H atom electron at r1

~H atom electron at r2

newly introduced

: Correlated, coupled

Hartree-Fock equation (One-electron equation)

spherically symmetric

&

- Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way.

- Any one electron sees only the spatially averaged position of all other electrons.

- VHF(i) is spherically symmetric.

- (Instantaneous, dynamic) electron correlation is ignored.

- Spherical harmonics (s, p, d, …) are validangular-part eigenfunctions (as for H-like atoms).

- Radial-part eigenfunctions of H-like atoms are not valid any more. optimized

Veff includes

• A single Slater determinant never corresponds to the exact wave function.

EHF > E0 (the exact ground state energy)

• Correlation energy: a measure of error introduced through the HF scheme

EC = E0 EHF (< 0)

– Dynamical correlation

– Non-dynamical (static) correlation

• Post-Hartree-Fock method (We’ll see later.)– Møller-Plesset perturbation: MP2, MP4, …

– Configuration interaction: CISD, QCISD, CCSD, QCISD(T), …

– Multi-configuration self-consistent-field method: MCSCF, CAFSCF, …

Electron Correlation (P.-O. Löwdin, 1955)Ref) F. Jensen, Introduction to Computational Chemistry, 2nd ed., Ch. 4

Solution of HF-SCF equation gives

Solution of HF-SCF equation:Z- (measure of shielding)0 0.31

1.72 2.09

8.49 8.69

2.422.58

2.782.86

3.153.17

3.513.55

3.873.90

4.244.24

8.888.93

9.109.71

9.3610.11

9.7310.52

9.9310.88

10.2411.24

more shieldedless shielded

Solution of HF-SCF equation:Effective nuclear charge

(Z- is a measure of shielding.)

higher energy, bigger radius lower energy, smaller radius

www.periodictable.com/Properties/A/AtomicRadius.v.wt.html

Source: www.chemix-chemistry-software.com/school/periodic_table/atomic-radius-elements.html

larger smaller

As well as the total energy, one also obtains a set of orbital energies.

Remove an electron from occupied orbital a.

Orbital energy = Approximate ionization energy

Physical significance of orbital energies (i):Koopmans’ theorem (T. C. Koopmans, 1934)

Physica,1, 104

Ostlund/SzaboCh.3.3

length

energy

Atomic orbital energy levels & Ionization energyof H-like atoms

2

20

0

4

ema

e

Total energy eigenvalues are negative by convention. (Bound states)

...3,2,1 with 32 222

02

42

nne

eZEn

depend only on the principal quantum

number.

1Ry

Minimum energy required to remove an electron from the ground state

IE (1 Ry for H)

atomic units

Koopmans’ theorem: Validation from experiments

Hartree-Fock orbital energies i & Aufbau principle

degenerate

For H-like atoms

”

”

Hartree-Fock orbital energies i depend on

both the principal quantum number (n) and the angular quantum number (l).

Within a shell of principal quantum number n,

ns np nd nf …

Aufbau (Building-up) principle for transition metals

10.3

Aufbau (Building-up) principle for transition metals

Electronegativity (~ IE + EA)

~Lowest Unoccupied

AO/MO (LUMO)

~Highest Occupied

AO/MO (HOMO)

small

small

high

high

low or deep

low or deep

small

large

large

large

Na + Cl+ NaCl Na+ + Cl

Periodic trends of many-

electron atoms

Periodic trends of many-electron atoms: Electronegativity

http://www.periodictable.com/Properties/A/Electronegativity.bt.wt.html

Periodic trends of many-electron atoms: 1st ionization energy

http://www.periodictable.com/Properties/A/IonizationEnergies.bt.wt.html

Periodic trends of many-electron atoms: Electron affinity

http://www.periodictable.com/Properties/A/ElectronAffinity.bt.wt.html

Periodic trends of many-electron atoms: “Atomic” radius

http://www.periodictable.com/Properties/A/AtomicRadius.bt.wt.html